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In this paper, we present a novel Brillouin fiber-ring laser utilizing an unbalanced Mach-Zehnder interferometer (UMZI) as coupling device. The laser is pumped by a distributed-feedback laser diode and shows continuous-wave and single-frequency operation. Frequency-dependent transmission characteristics of the UMZI make it possible for the pump wave to pass through the laser-ring cavity with no resonance effect for stable pump operation, while the Brillouin laser signal still resonates in a high-finesse cavity. Single and multiple longitudinal mode operations are observed according to the relative location between longitudinal modes and Brillouin gain-curve center. A stable single-frequency operation is achieved using a simple stabilizing feedback loop based on dithering and autotracking techniques. Using this simple stabilizing feedback loop, the laser-intensity fluctuation is highly suppressed and remains below 4%. The Brillouin output converted from the pump power of 26.4 mW is about 3.18 mW, and the linewidth is measured to be below 1 kHz.